1,999,440 research outputs found
Causal diffusion and its backwards diffusion problem
This article starts over the backwards diffusion problem by replacing the
\emph{noncausal} diffusion equation, the direct problem, by the \emph{causal}
diffusion model developed in \cite{Kow11} for the case of constant diffusion
speed. For this purpose we derive an analytic representation of the Green
function of causal diffusion in the wave vector-time space for arbitrary (wave
vector) dimension . We prove that the respective backwards diffusion problem
is ill-posed, but not exponentially ill-posed, if the data acquisition time is
larger than a characteristic time period () for space dimension
(N=2). In contrast to the noncausal case, the inverse problem is
well-posed for N=1. Moreover, we perform a theoretical and numerical comparison
between causal and noncausal diffusion in the \emph{space-time domain} and the
\emph{wave vector-time domain}. The paper is concluded with numerical
simulations of the backwards diffusion problem via the Landweber method.Comment: In the replacement I have rewritten the abstract and the
introduction. Moreover, I have added Remark 1 and simplified a little bit the
proof of Theorem 4. The reference 25 is updated, since the paper is now
publishe
Diffusion-Reorganized Aggregates: Attractors in Diffusion Processes?
A process based on particle evaporation, diffusion and redeposition is
applied iteratively to a two-dimensional object of arbitrary shape. The
evolution spontaneously transforms the object morphology, converging to
branched structures. Independently of initial geometry, the structures found
after long time present fractal geometry with a fractal dimension around 1.75.
The final morphology, which constantly evolves in time, can be considered as
the dynamic attractor of this evaporation-diffusion-redeposition operator. The
ensemble of these fractal shapes can be considered to be the {\em dynamical
equilibrium} geometry of a diffusion controlled self-transformation process.Comment: 4 pages, 5 figure
Steady states in a structured epidemic model with Wentzell boundary condition
We introduce a nonlinear structured population model with diffusion in the
state space. Individuals are structured with respect to a continuous variable
which represents a pathogen load. The class of uninfected individuals
constitutes a special compartment that carries mass, hence the model is
equipped with generalized Wentzell (or dynamic) boundary conditions. Our model
is intended to describe the spread of infection of a vertically transmitted
disease, for example Wolbachia in a mosquito population. Therefore the
(infinite dimensional) nonlinearity arises in the recruitment term. First we
establish global existence of solutions and the Principle of Linearised
Stability for our model. Then, in our main result, we formulate simple
conditions, which guarantee the existence of non-trivial steady states of the
model. Our method utilizes an operator theoretic framework combined with a
fixed point approach. Finally, in the last section we establish a sufficient
condition for the local asymptotic stability of the positive steady state
Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations
We propose diffusion-like equations with time and space fractional
derivatives of the distributed order for the kinetic description of anomalous
diffusion and relaxation phenomena, whose diffusion exponent varies with time
and which, correspondingly, can not be viewed as self-affine random processes
possessing a unique Hurst exponent. We prove the positivity of the solutions of
the proposed equations and establish the relation to the Continuous Time Random
Walk theory. We show that the distributed order time fractional diffusion
equation describes the sub-diffusion random process which is subordinated to
the Wiener process and whose diffusion exponent diminishes in time (retarding
sub-diffusion) leading to superslow diffusion, for which the square
displacement grows logarithmically in time. We also demonstrate that the
distributed order space fractional diffusion equation describes super-diffusion
phenomena when the diffusion exponent grows in time (accelerating
super-diffusion).Comment: 11 pages, LaTe
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